# 7 3 Triangle Similarity AA SSS SAS 7

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7 -3 Triangle. Similarity: AA, SSS, SAS 7 -3 Triangle Warm Up Lesson Presentation Lesson Quiz Holt Mc. Dougal Geometry

7 -3 Triangle Similarity: AA, SSS, SAS Warm Up Solve each proportion. 1. 2. 3. z = ± 10 x=8 4. If ∆QRS ~ ∆XYZ, identify the pairs of congruent angles and write 3 proportions using pairs of corresponding sides. Q X; R Y; S Z; Holt Mc. Dougal Geometry

7 -3 Triangle Similarity: AA, SSS, SAS Objectives Prove certain triangles are similar by using AA, SSS, and SAS. Use triangle similarity to solve problems. Holt Mc. Dougal Geometry

7 -3 Triangle Similarity: AA, SSS, SAS There are several ways to prove certain triangles are similar. The following postulate, as well as the SSS and SAS Similarity Theorems, will be used in proofs just as SSS, SAS, ASA, HL, and AAS were used to prove triangles congruent. Holt Mc. Dougal Geometry

7 -3 Triangle Similarity: AA, SSS, SAS Example 1: Using the AA Similarity Postulate Explain why the triangles are similar and write a similarity statement. Since , B E by the Alternate Interior Angles Theorem. Also, A D by the Right Angle Congruence Theorem. Therefore ∆ABC ~ ∆DEC by AA~. Holt Mc. Dougal Geometry

7 -3 Triangle Similarity: AA, SSS, SAS Holt Mc. Dougal Geometry

7 -3 Triangle Similarity: AA, SSS, SAS Holt Mc. Dougal Geometry

7 -3 Triangle Similarity: AA, SSS, SAS Example 2 A: Verifying Triangle Similarity Verify that the triangles are similar. ∆PQR and ∆STU Therefore ∆PQR ~ ∆STU by SSS ~. Holt Mc. Dougal Geometry

7 -3 Triangle Similarity: AA, SSS, SAS Example 2 B: Verifying Triangle Similarity Verify that the triangles are similar. ∆DEF and ∆HJK D H by the Definition of Congruent Angles. Therefore ∆DEF ~ ∆HJK by SAS ~. Holt Mc. Dougal Geometry

7 -3 Triangle Similarity: AA, SSS, SAS Example 3: Finding Lengths in Similar Triangles Explain why ∆ABE ~ ∆ACD, and then find CD. Step 1 Prove triangles are similar. A A by Reflexive Property of , and B C since they are both right angles. Therefore ∆ABE ~ ∆ACD by AA ~. Holt Mc. Dougal Geometry

7 -3 Triangle Similarity: AA, SSS, SAS Example 3 Continued Step 2 Find CD. Corr. sides are proportional. Seg. Add. Postulate. x(9) = 5(3 + 9) 9 x = 60 Substitute x for CD, 5 for BE, 3 for CB, and 9 for BA. Cross Products Prop. Simplify. Divide both sides by 9. Holt Mc. Dougal Geometry

7 -3 Triangle Similarity: AA, SSS, SAS Example 5: Engineering Application The photo shows a gable roof. AC || FG. ∆ABC ~ ∆FBG. Find BA to the nearest tenth of a foot. From p. 473, BF 4. 6 ft. BA = BF + FA 6. 3 + 17 23. 3 ft Therefore, BA = 23. 3 ft. Holt Mc. Dougal Geometry

7 -3 Triangle Similarity: AA, SSS, SAS Check It Out! Example 5 What if…? If AB = 4 x, AC = 5 x, and BF = 4, find FG. Corr. sides are proportional. Substitute given quantities. 4 x(FG) = 4(5 x) Cross Prod. Prop. FG = 5 Holt Mc. Dougal Geometry Simplify.

7 -3 Triangle Similarity: AA, SSS, SAS You learned in Chapter 2 that the Reflexive, Symmetric, and Transitive Properties of Equality have corresponding properties of congruence. These properties also hold true for similarity of triangles. Holt Mc. Dougal Geometry

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